To proceed further, we use determinant identities and the Laplace expansion to rewrite \(\sigma _n\)  in Eq. (REF ) as [1]}
\( && \hspace{-22.76228pt} \sigma _{n}=\sum _{0\le \nu _{1} < \nu _{2} < \cdots < \nu _{N}\le 2N-1}\det _{1 \le i, j\le N} \left[\frac{1}{2^{\nu _j}} S_{2i-1-\nu _j}(\textbf {\emph {x}}^{+}(n) +\nu _j \textbf {\emph {s}}) \right] \times \det _{1 \le i, j\le N}\left[\frac{1}{2^{\nu _j}}S_{2i-1-\nu _j}(\textbf {\emph {x}}^{-}(n) + \nu _j \textbf {\emph {s}})\right].\) 
